Adaptive complex gain predistorter for a transmitter

ABSTRACT

Symbols are transmitted in a Cartesian transmitter by pre-distorting an input signal X having in-phase and quadrature components using a first compensation lookup table operable to hold complex valued entries to carry out in-phase and quadrature compensation pre-distortion with respect to the input signal to form a pre-distorted signal Z. The pre-distorted signal Z is processed to form an output signal Y using a nonlinear element. A complex gain normalization parameter adaptively updated to reflect varying gain of a linear region of the nonlinear element. A normalized feed back signal {tilde over (Y)} is formed using the adaptively updated complex gain normalization parameter. The first compensation lookup table is updated based on the pre-distorted input signal Z and the adaptively normalized feedback signal {tilde over (Y)}.

CLAIM OF PRIORITY UNDER 35 U.S.C. 119(e)

The present application claims priority to and incorporates by referenceU.S. Provisional Application No. 61/086,261 filed Aug. 5, 2008, entitled“Adaptive Complex Gain Predistorter for a Handset Transmitter withAdaptive Gain/Phase Equalization”.

FIELD OF THE INVENTION

The invention is directed, in general, to wireless transmitters and,more specifically, to an apparatus and method for adaptive Cartesiantransmitter linearization and a wireless transmitter for use in cellulartelephony and communication devices such as Bluetooth, WLAN, etc. usingall-digital radio frequency (RF) circuitry.

BACKGROUND OF THE INVENTION

Wireless cellular communication networks incorporate large numbers ofmobile user equipment (UEs) and a number of base nodes (NodeBs). A NodeBis generally a fixed station, and may also be called a base transceiversystem (BTS), an access point (AP), a base station (BS), or some otherequivalent terminology. As improvements of networks are made, the NodeBfunctionality evolves, so a NodeB is sometimes also referred to as anevolved NodeB (eNB). In general, NodeB hardware, when deployed, is fixedand stationary, while the UE hardware is typically portable.

In contrast to NodeB, the mobile UE can comprise portable hardware. Userequipment (UE), also commonly referred to as a terminal or a mobilestation, may be fixed or mobile device and may be a wireless device, acellular phone, a personal digital assistant (PDA), a wireless modemcard, and so on. Uplink communication (UL) refers to a communicationfrom the mobile UE to the NodeB, whereas downlink (DL) refers tocommunication from the NodeB to the mobile UE. Each NodeB contains radiofrequency transmitter(s) and the receiver(s) used to communicatedirectly with the mobiles, which move freely around it. Similarly, eachmobile UE contains radio frequency transmitter(s) and the receiver(s)used to communicate directly with the NodeB. In cellular networks, themobiles cannot communicate directly with each other but have tocommunicate with the NodeB.

With each successive cellular phone handset generation, users demandmore features in a smaller form factor. Some recent examples includecell phones with integrated Bluetooth, GPS, digital camera, and MP3functionality. Process shrinks help deliver a cost and size advantagefor digital designs with relative ease. However, for analog/RF designs,the immaturity of advanced processes comes with design challenges thatmay outweigh the intended advantage. In a typical handset, 30 to 40% ofhandset board space is occupied by analog/RF functionality which cannotbe re-designed or migrated to the newer process/technology nodes easily,inhibiting vendor ability to cost effectively add features and reducefootprint.

Digital radio has recently allowed the replacement of space consuminganalog RF circuitry with much more compact digital circuitry, therebyfacilitating the ability to port designs rapidly to more advancedlithographies. Texas Instruments (TI) has proven this concept with itsDigital RF Processor (DRP™) architecture, which it has successfullyimplemented in production versions of its Bluetooth BRF6xxxtransceivers, GSM/GPRS LoCosto TCS23xx transceivers among other chips.DRP implementation is consistent with the on-going trend toward RF-CMOSin the cellular area, making it attractive in terms of powerconsumption, cost, and the integration of multiple radios.

Transmitters use one or more amplifiers, such as a digital pre-poweramplifier (PPA) and an external power amplifier (PA), to amplifycomponents of the input signal to be transmitted. These components arein-phase and quadrature components in the case of a Cartesiantransmitter.

A highly linear amplifier distorts the signal the least and so is mostfavored from a standpoint of signal quality. Unfortunately, highlylinear amplifiers use relatively large amounts of power and numbers ofhighly accurate and tightly matched components, making them relativelypower consumptive, large and expensive. Though they perform the best,they are thus disfavored in many wireless applications, particularlythose that require low-cost transmitters or transmitters that aresubject to large operating voltage excursions. The amplifier that isbest suited overall for low-cost, battery-powered wireless transmittersis a simpler amplifier having significant nonlinearities.

Predistortion is often used to compensate for these nonlinearities,resulting in a linearization of the output of the amplifier. The theoryunderlying predistortion is that if an amplifier's distortioncharacteristics are known in advance, an inverse function can be appliedto an input signal to predistort it before it is provided to theamplifier. Though the amplifier then distorts the signal as it amplifiesit, the predistortion and the amplifier distortion essentially cancelone another, resulting in an amplified output signal havingsubstantially reduced distortion.

In digital transmitters, digital predistortion (DPD) is most oftencarried out using a lookup table (LUT) that associates output valueswith input signal values. Entries in the LUT are addressed using samplesof the input signal. The output values retrieved from the LUT are usedeither to modify the samples (an “inverse gain” configuration) or inlieu of the samples (a “direct mapping” configuration). In modernapplications such as WCDMA, samples are transmitted at a very high rate.Thus, the predistorter needs to be able to look up and retrieve outputvalues very quickly.

WCDMA Cartesian transmitters suffer nonlinearities resulting from bothamplitude modulation (AM) and phase modulation (PM), namely AM-AM andAM-PM interactions, occurring in their amplifier(s). In such Cartesiantransmitters, predistortion is carried out at least partially to negatethe effect of these interactions.

Values for a nominal predistortion LUT are typically computed duringinitial factory calibration. Unfortunately, a factory-calibratedpredistortion LUT often fails to linearize the amplifier(s) adequatelyunder varying operational conditions (e.g., temperature, voltage,frequency and voltage standing-wave ratio, or VSWR). Aging, especiallyin WCDMA and other so-called “3G” transmitters, only exacerbates theinadequacy.

BRIEF DESCRIPTION OF THE DRAWINGS

Particular embodiments in accordance with the invention will now bedescribed, by way of example only, and with reference to theaccompanying drawings:

FIG. 1 is a pictorial of an illustrative telecommunications network thatemploys an embodiment of adaptive gain normalization and predistortionin transceivers used in the network;

FIG. 2 is a block diagram of a single-chip radio with an all-digitallocal oscillator and transmitter that performs predistortion;

FIGS. 3A and 3B are plots illustrating nonlinear operation of thetransmitter of FIG. 2;

FIGS. 4A and 4B are plots illustrating typical Class A and Class C,respectively, operation of the transmitter amplifier of FIG. 2;

FIG. 5 is a more detailed block diagram of the transmitter portion ofthe radio of FIG. 2 that embodies adaptive gain normalization andpredistortion;

FIG. 6 is a more detailed block diagram of the adaptive predistortionmechanism of the transmitter of FIG. 5;

FIGS. 7A and 7B are plots illustrating entries in a complex look uptable used in the predistortion mechanism of FIG. 6;

FIG. 8 is a block diagram illustrating the adaptive gain/phasenormalization mechanism of FIG. 6;

FIG. 9 is a flow diagram illustrating operation of the adaptivenormalization and predistortion;

FIGS. 10A and 10B are plots comparing EVM and ACLR1, respectively,performance with adaptive gain/phase normalization;

FIG. 11 is a block diagram illustrating a module to determineapproximate amplitude for indexing the look up tables of FIG. 6; and

FIG. 12 is a block diagram of a cellular telephone with an embodiment ofa transmitter using adaptive normalization and predistortion.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

Embodiments of the present invention include adaptive digitallinearization techniques for a Cartesian WCDMA transmitter. A digitalPredistortion block implemented as a digital lookup table (LUT) is usedto predistort the complex baseband signal (Inphase and Quadraturecomponents) of a WCDMA Cartesian transmitter. The transmitternon-linearity is assumed to be dominant in the post combination devicessuch as Pre-power Amplifier (PPA) and/or the external Power amplifier.As result the dominant nonlinearity will be a function of the complexmagnitude of the baseband signal as opposed to the magnitude of Inphaseand Quadrature components (such as is the case for the DAC and mixernonlinearity). These non-linearities distort the I/Q signalconstellation that is transmitted by the TX, thus causing poor errorvector magnitude (EVM) and spectral degradation, which is observed as adegradation of the adjacent channel leakage power/ratio (ACLP/ACLR) andpossibly increased broadband noise.

To aid in understanding the principles of the present invention, adescription is provided in the context of a digital RF processor (DRP)transmitter and receiver that may be adapted to comply with a particularwireless communications standard such as GSM (Global System for Mobilecommunication), Bluetooth, WCDMA (Wideband Code Division MultipleAccess), etc. It is appreciated, however, that the invention is notlimited to use with any particular communication standard and may beused in control, optical, wired and wireless applications. Further, theinvention is not limited to use with a specific modulation scheme but isapplicable to any modulation scheme including both digital and analogmodulation.

Note that throughout this document, the term communications device isdefined as any apparatus or mechanism adapted to transmit, or transmitand receive data through a medium. The communications device may beadapted to communicate over any suitable medium such as RF, wireless,infrared, optical, wired, microwave, etc. In the case of wirelesscommunications, the communications device may comprise an RFtransmitter, RF receiver, RF transceiver or any combination thereof. Thenotation DRP is intended to denote either a Digital RF Processor orDigital Radio Processor. References to a Digital RF Processor infer areference to a Digital Radio Processor and vice versa.

In an embodiment of the invention, a least means squared (LMS) basedadaptation technique may be used for a complex gain lookup table (LUT)to track the temperature variations of the transmitter's nonlinearcharacteristics during operation. The algorithm is optimized to enable alow complexity hardware implementation and provides relatively fastconvergence. An amplitude approximation method that is suitable toaddress the complex gain LUT is described that avoids the computation ofsquare root. Also, an adaptive loop gain/phase normalization techniqueis described that avoids the appearance of curve discontinuities andmaintains the accuracy of open loop power control. It reuses theexisting LUT adaptation hardware and therefore is inexpensive toimplement.

FIG. 1 shows an exemplary wireless telecommunications network 100. Theillustrative telecommunications network includes representative basestations 101, 102, and 103; however, a telecommunications networknecessarily includes many more base stations. Each of base stations 101,102, and 103 are operable over corresponding coverage areas 104, 105,and 106. Each base station's coverage area is further divided intocells. In the illustrated network, each base station's coverage area isdivided into three cells. Handset or other UE 109 is shown in Cell A108, which is within coverage area 104 of base station 101. Base station101 is transmitting to and receiving transmissions from UE 109 viadownlink 110 and uplink 111. As UE 109 moves out of Cell A 108, and intoCell B 107, UE 109 may be handed over to base station 102. A UE in acell may be stationary such as within a home or office, or may be movingwhile a user is walking or riding in a vehicle. UE 109 moves within cell108 with a velocity 112 relative to base station 102.

In one embodiment of the invention, UE 109 is transmitting to andreceiving from base station 101 voice and/or data transmissions. As theUE moves around in the network and is placed in different environmentssuch as in a pocket or purse, carried to hot or cold environments suchas indoors and outdoors, placed in the sun, and so on, the resultingextreme changes in temperature will cause differences in the performanceof analog components used in the transmitter. An adaptive loopgain/phase normalization mechanism along with an adaptive predistorteris used to compensate for these temperature variations, as will beexplained in more detail below.

A block diagram illustrating a single chip radio incorporating aninterpolative all-digital local oscillator based Cartesian transmitterand digitally-intensive receiver is shown in FIG. 2. For illustrationpurposes only, the transmitter, as shown, is adapted for the EDGE/WCDMAcellular standards. It is appreciated, however, that one skilled in thecommunication arts can adapt the transmitter illustrated herein to othermodulations and communication standards as well without departing fromthe spirit and scope of the present invention. This embodiment of a DRPufor UMTS is a Digital RF Processor (DRP)-based dominantly digitaltransceiver integrated with a digital baseband processor in 45 nm CMOStechnology. This DRPu EDGE/WCDMA (2.5G/3G) transmitter (TX) is based ona Cartesian (I/Q) direct up-conversion TX architecture with digitalassistance for calibrations and compensation, henceforth termed as aDigitally Assisted analog I/Q (DAIQ) TX. For GSM (2G), the transmitterarchitecture is small-signal analog polar. DRP supports interface withboth multi-mode and multi-band power amplifiers (PA).

The radio circuit, generally referenced 130, comprises a transceiverintegrated circuit (IC) 136 coupled to a crystal 152, antenna front endmodule 176 connected to antenna 180 and battery management circuit 132.The radio chip 136 comprises a script processor 146, memory 142 (e.g.,static RAM), transmit (TX) block 148, receiver (RX) block 150, digitallycontrolled crystal oscillator (DCXO) 154, slicer 156, power managementunit 138, RF built-in self test (BIST) 140. Battery 134 and batterymanagement circuit 132 are connected to radio chip 136 for providingpower. Digital baseband (DBB) processor 144 and flash memory/EEPROM 145is coupled to transceiver IC 136 via transceiver interface 137.

The TX block 148 comprises high speed and low speed digital logic block158, digital to analog converter 160, low pass filter 162, amplitudemodulator 168, digitally controlled oscillator (DCO) 164, digitallycontrolled pre-power amplifier 174. The transmitter generates variousradio frequency signals, as defined by the 3GPP specifications. Forexample, the transmitter may support one or more of the 3 G UMTSfrequencies: 850, 900, 1700, 1900, or 2100 MHz.

A key component of transmitter block 148 is digitally controlledoscillator (DCO) 164, that is part of an interpolated-digitalphase-locked loop (ADPLL). DCO 164 avoids any analog tuning controls.The DCO generates a high-quality base station-synchronized frequencyreference such that the transmitted carrier frequencies and the receivedsymbol rates are accurate to within 0.1 ppm. Fine frequency resolutionis achieved through high-speed sigma-delta (ΣΔ) dithering of itsvaractors. Digital logic built around the DCO realizes an interpolatedall-digital PLL (ADPLL) that is used as a local oscillator for both thetransmitter and receiver. The Cartesian transmitter architectureutilizes a digitally controlled power amplifier (DPA) 174 for theamplitude modulation. It is followed by a matching network and anexternal antenna front-end module 176, which comprises a power amplifier(PA), a transmit/receive switch for the common antenna 180 and RXsurface acoustic wave (SAW) filters.

An advanced All-Digital PLL (ADPLL) frequency synthesizer is describedin US Patent application 2008-0315960 to Waheed et al entitled “DigitalPhase Locked Loop with Gear Shifting” which is incorporated by referenceherein in its entirety.

Fixed baseband clock circuit 155 provides a fixed clock to DBB processor144 and to transceiver interface 137. Clock module 166 receives avariable clock from DCO 164 and produces a set of synchronized RFderived clocks for use by digital processing module 158 and DAC 160.Clock module 166 also receives a clock signal from transceiver interface137 that is used to allow synchronization of clocks between fixed clock155 and variable clocks derived from DCO 164.

The receiver employs a discrete-time architecture in which the RF signalis directly sampled and processed using analog and digital signalprocessing techniques. RX block 150 comprises a low noisetransconductance amplifier 182, current sampler 184, discrete timeprocessing block 186, analog to digital converter (ADC) 188 and digitallogic block 190. The receiver 150 employs a discrete-time architecturein which the RF signal is directly sampled at the Nyquist rate of the RFcarrier and processed using analog and digital signal processingtechniques. The transceiver is integrated with a script processor 146,dedicated digital base band processor 144 (i.e. ARM family processor andDSP) and SRAM memory 142. The script processor handles various TX and RXcalibration, compensation, sequencing and lower-rate data path tasks andencapsulates the transceiver complexity in order to present a muchsimpler software programming model.

The frequency reference (FREF) is generated on-chip by a 38.4 MHz (butcould be 26.0 MHz or another frequency in another embodiment) digitallycontrolled crystal oscillator (DCXO). An integrated power management(PM) system is connected to an external battery management circuit 132that conditions and stabilizes the supply voltage. The PM comprises aswitched mode power supply (SMPS) as well as multiple low drop out (LDO)regulators that provide internal supply voltages and also isolate supplynoise between circuits, especially protecting the DCO. The SMPS is usedfor efficient conversion of the battery voltage to a level that can beused by on-chip LDOs. The RF built-in self-test (RFBIST) 140 performsautonomous phase noise and modulation distortion testing, variousloopback configurations for bit-error rate measurements and implementsvarious DPA calibration and BIST procedures.

In another embodiment, the transceiver may be integrated with thedigital baseband processor 144 and flash memory 145 memory in a completesystem-on-chip (SoC) solution.

FIGS. 3A and 3B are plots illustrating nonlinear operation of thetransmitter of FIG. 2. FIG. 3A illustrates in general how nonlinearamplifier 174, which is representative of PPA 174 in FIG. 2, distorts asubstantially sinusoidal input signal. In FIG. 3B, predistortion module302 is used to compensate for these nonlinearities, resulting in alinearization of the output of the amplifier. The theory underlyingpredistortion is that if an amplifier's distortion characteristics areknown in advance, an inverse function can be applied to an input signalto predistort it before it is provided to the amplifier. Though theamplifier then distorts the signal as it amplifies it, the predistortionand the amplifier distortion essentially cancel one another, resultingin an amplified output signal having substantially reduced distortion inwhich a digital predistorter 302 predistorts the substantiallysinusoidal input signal such that the output signal is likewisesinusoidal.

Predistorter 302 is embodied as a look-up table (LUT) that is indexed bythe input signal. At each instant of the input signal, the LUT isindexed by a representation of the input signal and the output of theLUT scales the input signal in response to the value at the indexedlocation in the LUT. In order to compensate for changes in temperature,Predistorter 302 needs to be continually updated. Existing complex gainLUT predistorters with sample-based update do not address the potentialdiscontinuities due to the local nature of the LUT update. Thesediscontinuities appear when the temperature changes greatly and onlypart of the LUT is updated. This highly undesirable behavior in overcomeby adaptive gain/phase normalization as will be described in more detailbelow.

FIGS. 4A and 4B are plots illustrating typical Class A and Class C,respectively, operation of the transmitter amplifier of FIG. 2. Thenonlinearity of the PPA/PA distorts the combined RF signal. Thedistortion is caused by the amplitude depend gain and phase shift of thePPA/PA. the distortion is modeled by a complex gain F(r). The AM-AM andAM-PM are respectively given by the amplitude dependant magnitude andangle of F(r). Examples of AM-AM and AM-PM curves are shown FIGS. 4A and4B.

FIG. 5 is a more detailed block diagram of the transmitter portion ofthe radio of FIG. 2 that embodies adaptive gain normalization andpredistortion in a Cartesian transmitter. Digital baseband processor 144(illustrated in FIG. 2) produces in-phase and quadrature components Iand Q. A filter block 505 conditions I and Q for amplification.Cartesian predistortion LUT 510 is a single LUT that is initially loadedwith calibration based entries and then is updated adaptively. It isemployed to produce predistorted amplitude and phase components I_(Z)and Q_(Z). I_(Z) and Q_(Z) are converted to analog form, filtered andmodulated as shown and provided to the PPA 535, yielding the WCDMAoutput signal. I_(Z) and Q_(Z) are also provided to the decimator andaligner 545 to be used in temperature adaptation. Phase locked loop 520is representative of the ADPLL in FIG. 2 that includes digitallycontrolled oscillator 164.

A coupler 540 provides a portion of the WCDMA output signal to the inputof a receiver. In this embodiment coupler 540 is a signal trace on thecircuit substrate. The receiver employs a low noise amplifier (LNA) 550to yield RF signals that are then down-converted to produce in-phase andquadrature components I_(Y) and Q_(Y) of the WCDMA output signal, whichare then filtered in low pass filters (LPF) as shown. A second Cartesianpredistortion LUT 560 also has one LUT that holds complex valued entriesin this embodiment. LUT 560 post-distorts I_(Z) and Q_(Z). Thedifferences between these amplitude and phase components and thoseprovided via the decimator and aligner 645 are provided to apredistortion adapter 665 which updates predistortion in the complexcompensation LUT of the second Cartesian predistortion LUT 560. Thesecond Cartesian predistortion LUT 560 are then exchanged with the firstCartesian predistortion LUTs 510 for the next lookup. The firstCartesian predistortion LUTs 510 are updated during that next lookup,the first and second Cartesian predistortion LUTs 510, 560 are exchangedagain for the lookup after that, and so on.

The feedforward predistortion may be either a direct mapping in whichthe value of an incoming complex signal determines the complexpredistorted value, or a complex scaling term in which the value of anincoming complex signal is scaled by the complex value in the LUT.

The Cartesian transmitter of FIG. 5 features three potentially nonlinearelements in both of its I and Q paths. These elements include thedigital-to-analog converter (DAC) 512, the PPA 514, possibly includingAM-AM and AM-PM nonlinearities, and PA 535, again possibly includingAM-AM and AM-PM nonlinearities.

FIG. 6 is a more detailed block diagram of the adaptive predistortionmechanism of the transmitter of FIG. 5. The distortion is modeled by acomplex gain F(r). The AM-AM and AM-PM are respectively given by theamplitude dependant magnitude and angle of F(r).

An input signal X is provided to predistortion LUT 604 that containscomplex predistortion scaling parameters. The complex output of LUT 604and the complex input signal X are multiplied in a scaling module 606and provided as input to a nonlinear element 608, which provides anoutput signal Y. A receiver feeds back the output signal Y, incurringsome delay. An output delay circuit 610 represents that delay. Thissignal is then normalized by normalization block 612 that multiples thefeedback signal by a complex gain normalization parameter that isroughly equivalent to an inverse of the gain of the front endcomponents. An input delay circuit 624 delays the output of scalingmodule 606 by an equivalent amount.

A second compensation LUT 618 receives the normalized feedback signal{tilde over (Y)}. The complex output of the second compensation LUT ismultiplied with the feedback signal {tilde over (Y)} in scaling module620. A summing junction 622 provides the difference signal ε_(k) betweenthe two delayed signals to a quality monitor 626. The quality monitoruses an iterative LMS-like approach to update the second compensationLUT 618 to minimize the difference. The exact method of performing theupdate will be described in more detail below. The compensationpredistorter is adapted during transmission using the symbols that arebeing transmitted. The compensation predistorter continuously changesuntil a steady state is reached when the feedback error reaches zero.After an entry in second LUT 618 is updated, multiplexing logic thatcontrols the LUTs is toggled so that the feed-forward predistortionpoints to the newly updated table and the adaptation points to the otherLUT, as indicated at 630.

Adaptive gain/phase equalization module 614 updates normalization block612 to compensate for changes in loop gain due to temperature andvoltage variations that affect the operation of analog components of thetransmitter. The transmitter's nonlinearity varies over time because oftemperature fluctuations, but also fast varying loading conditions(VSWR). Inaccuracies in the gain normalization can cause thepredistorter curves to vary drastically. Since the update is local thispotentially causes temporary discontinuities in the LUT curves thatcould degrade the transmitter's performance. An adaptive gainnormalization that reuses the predistorter update hardware mitigatesthis issue.

The adaptive normalization mechanism is operable to temporarilysubstitute a single register for the second compensation lookup table618 while updating the complex gain normalization parameter. As will bedescribed in more detail below, the adaptive normalization mechanismiteratively converges on a revised gain normalization parameter usingthe single register to determine a complex value equal to approximatelythe inverse of a combined gain of the non-linear device over a linearregion and components of the feedback loop, such that a resulting loopgain is approximately one.

First LUT 604 is indexed using a representation of the input signal X.Amplitude computation through the formula√{square root over (I ² +Q ²)}

has a high computational complexity. As a result most the practicalimplementations use the squared amplitude instead]. In this disclosurean amplitude approximation suitable for the complex gain predistorterconfiguration is used. This approximation allows very efficient hardwareimplementation and has even lower computational complexity than thesquared amplitude indexing. Approximation module 602 determines anapproximate value corresponding to each sample of input signal X that isused to index first LUT 604. Similarly, approximation module 616determines an approximate value corresponding to each sample of feedbacksignal {tilde over (Y)} that is used to index second LUT 618. Theoperation of the approximation modules will be described in more detailbelow.

As described above, the predistorter is realized with a lookup tablestoring the imaginary and real parts of the complex gain. Thetransmitter is perfectly linearized when the following condition issatisfied:

X × G[m] × F(X × G[m]) = K × X ⇒ G[m] × F(X × G[m]) = KThe parameter K is the desired gain of the linearized transmitter.

The present adaptation technique using a feedback path can be used forboth factory self-calibration and online update. A low complexity updateequation is obtained by applying the least mean squares (LMS) algorithm.The error ε_(k) produced by error detection module 622 is given by:ε_(k) =Z−LUT[n]×{tilde over (Y)}

The goal is to minimize an objective function which is the mean squarederror:E(|ε_(k)|²)≈(Z−{tilde over (Y)}×LUT[n])×(Z−{tilde over (Y)}×LUT[n])*

The update equation is then given by:

$\begin{matrix}\begin{matrix}{{L\; U\;{T\lbrack n\rbrack}} = {{L\; U\;{T\lbrack n\rbrack}} - {\frac{\mu}{2} \times \frac{{\partial E}\lfloor {ɛ_{k}}^{2} \rfloor}{{\partial L}\; U\;{T\lbrack n\rbrack}}}}} \\{\approx {{L\; U\;{T\lbrack n\rbrack}} + {\mu \times {\overset{\sim}{Y}}^{*} \times ɛ_{k}}}}\end{matrix} & (1)\end{matrix}$where:LUT[n] indicates an entry in the second compensation lookup table,{tilde over (Y)}* is a complex conjugate value of the normalizedfeedback signal {tilde over (Y)}.ε_(k) is an error signal corresponding to a difference betweenpre-distorted signal Z and post-distorted signal Zfb; andμ is an update factor (real-valued) that controls the speed ofconvergence.

The above update equations (1) require only two complex multipliers, onefor the computation of the error and one for update, and two complexadders. A prior scheme required four complex multipliers and a divisionand is therefore unsuitable to low cost cellular handset design.

FIGS. 7A and 7B are plots illustrating entries in a complex look uptable used in the predistortion mechanism of FIG. 6. These entries aredetermined by quality monitor 626 performing the update operationdescribed above.

Low Hardware Complexity Update Calculation

The feed forward signal Z (Z=I_(p)+jQ_(p)) and feedback signal Y(Y=I_(o)+jQ_(o)) signals are fed to the error detection block 622 whichcomputes a complex error value (I and Q paths) based on these inputs.This error is used to calculate the updated LUT 618 entries as discussedabove. The error is given by:ε_(k) =Z−LUT[n]×Yε_(k)=(I _(p) +jQ _(p))−LUT[n]×(I _(o) +jQ _(o))

The complex gain LUT[n], also referred to as G(n), is obtained bylooking up an entry in the LUT indexed by a representative value offeedback signal Y. A nearest neighbor LUT is considered here, but theoptimization presented here is also applicable to linearly interpolationand higher orders of interpolation.

From the previously derived update equation, the LUT is updated asfollows:LUT[n]=LUT[n]+δ _(G)

where δ_(G) is given by:δ_(G)=ε_(k)×(I _(o) −jQ _(o))×μ_(k)

The above equation should be normalized by the amplitude of the feedbacksignal to improve the update speed at low amplitude levels:

$\delta_{G} = {ɛ_{k} \times \frac{( {I_{o} - {j\; Q_{o}}} )}{\sqrt{I_{o}^{2} + Q_{o}^{2}}} \times \mu_{k}}$$\delta_{G} = {ɛ_{k} \times {\mathbb{e}}^{{- j}\;{arc}\;{\tan{(\frac{Q_{o}}{Io})}}} \times \mu_{k}}$

It can be seen from this equation that the complex update (δ_(G))calculation requires the rotation of the error vector ε_(k) by theopposite of the feedback signal's angle. That is achieved above bymultiplying ε_(k) with the complex conjugate of the baseband feedbacksignal (I_(o)+jQ_(o)) and normalizing by the feedback amplitude. Thisrotation shown above can be simplified and avoid the complex multiplyand amplitude normalization by replacing

$\frac{( {I_{o} - {j\; Q_{o}}} )}{\sqrt{I_{o}^{2} + Q_{o}^{2}}}\mspace{14mu}{with}\mspace{14mu}{( {{{sign}( I_{o} )} - {j \cdot {{sign}( Q_{o} )}}} ).}$

This results in a “quantized” phase rotation by

${- k} \times \frac{\pi}{4}$where k is the quadrant number containing the vector (I_(o)+jQ_(o)).

$\begin{matrix}{\delta_{G} = {ɛ_{k} \times ( {{{sign}( I_{o} )} - {j \cdot {{sign}( Q_{o} )}}} ) \times \mu}} \\{= {ɛ_{k} \times ( {{\pm 1} \pm j} ) \times \mu}} \\{= {ɛ_{k} \times \sqrt{2 \times}{\mathbb{e}}^{{- j}\; k\frac{\pi}{4}} \times \mu}}\end{matrix}$

This new operation can be implemented very efficiently in hardwarewithout requiring multipliers. Only two additions/subtractions arereally needed. In addition to that, since amplitude normalization is notrequired the update factor does not need to be amplitude dependent andcan be maintained constant. Extensive simulations have shown this methodresults in exactly equal performance when compared to previousapproaches, even with exaggerated AM-AM and AM-PM characteristics.

Update Calculation for Linearly Interpolated LUT

In the case of the linearly interpolated LUT, two consecutive entriesare interpolated to provide the complex gain factor. In this case, theerror is given by:ε_(k)=(I _(p) +jQ _(p))−LUT_(i)×(I _(o) +jQ _(o))Where LUTi is given by:LUT_(i)=LUT[n]+interp_factor×(LUT[n+1]−LUT[n])

Since the LUT output is computed by linearly interpolating two entries,both of them will be simultaneously updated. The update to each of thetwo entries will be proportional to its contribution to the output asdescribed by the following equation:LUT[n]=LUT[n]+(1−interp_factor)×δ_(G)LUT[n+1]=LUT[n+1]+interp_factor×δ_(G)

As in the single entry case, the low complexity δ_(G) is given by:

$\begin{matrix}{\delta_{G} = {ɛ_{k} \times ( {{{sign}( I_{o} )} - {j \cdot {{sign}( Q_{o} )}}} ) \times \mu}} \\{= {ɛ_{k} \times ( {{\pm 1} \pm j} ) \times \mu}} \\{= {ɛ_{k} \times \sqrt{2 \times}{\mathbb{e}}^{{- j}\; k\frac{\pi}{4}} \times \mu}}\end{matrix}$where k is the quadrant number containing a vector (I_(o)+jQ_(o))corresponding to normalized feedback signal {tilde over (Y)}, andμ is an update factor (real-valued) that controls the speed ofconvergence.Adaptive Loop Gain Normalization

The predistortion update mechanism is very sensitive to loop gainnormalization variations. It must therefore be very accurate. Theinitial computed normalization 1/K used in normalization module 612,which is the product of the TX gain, the coupling factor and the LNAgain, for normalization is only an estimate and is not expected to bevery accurate, due to intractable temperature variations etc. . . .Therefore an adaptive gain normalization block 614 is used to improvethe normalization accuracy.

An adaptive loop gain normalization block 614 is introduced for thefollowing reasons:

-   -   The system can only provide an estimate of the overall loop gain    -   The gains of different components in feedback loop vary with        temperature. The predistorter adaptation will try to compensate        for the linear gain variations (as opposed to just nonlinear        components), causing the LUT curve to move up and down. This        will create curve discontinuities since the updates are only        local.    -   In addition to creating discontinuities when the linear        component of the predistortion curve varies, the accuracy of        open-loop power control will be severely degraded.    -   A complex adaptive gain normalization block is chosen to also        remove phase shifts that could be due to changes in matching        conditions etc. . . . Those phase shifts could also cause        temporary discontinuities in real and imaginary LUT curves        because of the local nature of the update.

The computation of the adaptive normalization is illustrated in FIG. 8.In FIG. 8, adaptive loop gain normalization block 614 is roughlyrepresented by adaptive normalization mechanism 806. Forwardcompensation block 802 represents LUT 604 and related circuitry.Adaptation module 804 represents LUT 618 and related circuitry. Acomplex factor α=α_(I)+jα_(Q) in register 808 is adapted such thatI _(p) +jQ _(p)=α×(I _(o) +jQ _(o)),over the linear code region only, such as region 402 or 404 in FIG. 4Aor 4B respectively. For example, linear regions 402, 404 are in a rangeof approximately 12 dB back-off or more. After convergence of 1/K,normalization is then updated by multiplying it with α and an additionalreal scaling factor β. This update of 1/K can be handled by the scriptprocessor. The update to α is applied only when (I_(p)+jQ_(p)) falls inthe linear region, 402 or 404 for example. A simple implementationconsists in testing if both |I_(p)| and |Q_(p)| are smaller than athreshold. The threshold should be programmable and could be set to apower of 2 to further reduce the implementation complexity. It isassumed that the AM-PM variations are very small in the linear region.In case of larger variations the update coefficient can be reduced toprovide further averaging.

The update equation for α is similar to the predistorter update. The LUTis simply replaced by the register 808 containing α. This can be donewith a multiplexor or other selection logic. First the error iscomputed:e _(k)=(I _(p) +jQ _(p))−α_(k)*(I _(o) +jQ _(o))where (I_(p)+jQ_(p)) is the feed forward signal and (I_(o)+jQ_(o)) isthe feedback signal.

Then update is applied by update logic 626:e _(k+1) =e _(k)+μ×(I _(O) −jQ _(O))where μ is an update factor (real-valued) that controls the speed ofconvergence.

Therefore the same hardware can be reused since the predistorter updateshould be activated only after correct loop normalization is done. Theonly additional function is the required testing to determine if thesignal is in the linear region. This is implemented very efficiently ifthe chosen threshold is a power of 2, in which case update logic 626only needs to test a single bit. It should be noted that the convergenceof α is several orders of magnitude faster than the predistortion LUTconvergence since all qualifying samples are used to update a singleregister as opposed to several entries of the LUT.

After convergence, a correction by a real factor β must be performed.This factor β must be greater than or equal to the maximum compressionat peak code, over all operating conditions (process, voltage,temperature, VSWR etc. . . . ). It is used to make sure that the peaksignal amplitude at the output of the predistorter is equal or less thanthe input's, so that the same bit representation is maintained at bothinput and output ports. The magnitude of the LUT gain will therefore bemaintained less than 1. If this additional correction is skipped thenthe predistorter will expand the range of the feed forward predistortionoutput signal potentially causing overflows unless the bit width of theoutput is adequately set.

After convergence, the normalization factors α and 1/K, and the realcorrection factor β are combined 810 into a single complex multiplierand stored in the 1/K register 812 by the script processor prior toactivating predistortion update.

A flow chart of the adaptive gain normalization is shown in FIG. 9.Periodically an adaptive gain/phase normalization operation isperformed. To begin, an estimated 1/K value is selected 902 and placedin 1/K register 812. α register 808 is initialized to 1+j0. Selectionmultiplexers are configured 904 to select α register 808 in place ofpredistortion adjustment LUT 618. A time counter is then initialized906.

Samples of complex input signal Z are tested 908 to determine if theyare in the linear region of nonlinear device 608. If yes, then αregister 808 is updated 910; otherwise the time count is incremented 908and the linearity test repeated on the next sample of input signal Z.This process repeats until α and 1/K converge 912 or until the timecounter expires. Typically, convergence occurs in less than 10 us. Atthis point, α register 808 and 1/K normalization register 812 aremultiplied together and scaled 914 by a predetermined constant β valuein order to guarantee the magnitude of the LUT gain will be maintainedless than 1. The multiplexers are then reset to select 916 predistortionadjustment LUT 618. Predistortion updates are then performed for aperiod of time until the adaptive normalization process is thenrepeated.

FIGS. 10A and 10B are plots comparing EVM and ACLR1, respectively,performance with adaptive gain/phase normalization as opposed to staticnormalization. As illustrated, gain variation up to +/−20 dB areaccommodated, with an accuracy better than 0.1 dB. Phase correctionaccuracy across the dynamic range is better than 5 degrees.

Efficient Amplitude Approximation

The exact computation of the amplitude from I/Q signals requires asquare root computation which is unpractical at high data processingrate (30.72 MHz). Therefore an approximation with less complexity isused.

The approximation works by first rotating the complex number so that itsangle is between 0 and π/4. This is simply done by calculating new realand imaginary parts (X and Y respectively) as follows:X=max(|I|,|Q|)Y=min(|I|,|Q|)

Then the amplitude is simply approximated by a linear combination of Xand Y:

$R = \{ \begin{matrix}{{a_{1}X} + {b_{1}Y}} & {0 \leq \theta < \theta_{B}} \\{{a_{2}X} + {b_{2}Y}} & {\theta_{B} \leq \theta < \frac{\pi}{4}}\end{matrix} $where θ is the angle of X+jY, which is between 0 and π/4.

Several exemplary sets of parameters a_(x), b_(x) and θ_(B) are definedin Table 1 for different options with increasing implementationcomplexity. The table gives tan(θ_(B)). Since tan ( ) is a strictlymonotonic increasing function, comparing two angles is the same ascomparing their tangents. Therefore, since the tangent is given asfraction tan(θ_(B))=c/d, comparing θ and θ_(B) is equivalent tocomparing d×Y and c×X.

TABLE 1 Parameters for 13 linear magnitude approximations Method# a₁ b₁tan (θ_(B)) a₂ b₂ 1 1 0 1 — — 2 1 1 1 — — 3 1 1/2 1 — — 4 1 1/4 1 — — 51 3/8 1 — — 6 31/32 3/8 1 — — 7 1 0 1/4 7/8 1/2 8 1 1/4 1/2 3/4 3/4 9 11/8 11/29 53/64 37/64 10 0.94800 0.39300 1 — — 11 0.96043 0.39782 1 — —12 0.98644 0.23287 1/2 0.81651 0.58851 13 0.99030 0.19698 tan (π/8)0.83954 0.56094

From Table 1, method 1 to method 9 can be implemented without amultiply, using only shifters and adders. Methods 10 to 13 requiremultiplications.

The above result suggests that on methods 6, 7 and 9 are preferable.These methods do not require multiplies and achieve relatively goodapproximations of the amplitude. The impact of these amplitudeapproximations on the performance of the feed-forward predistortion andadaptation process was simulated with Simulink test bench. All threemethods meet the specifications across all the performance parametersconsidered. Method 9 provides a far better approximation with aperformance very close to the ideal amplitude computation. Thereforemethod 9 provides the best trade-off between accuracy and implementationcomplexity.

FIG. 11 is a block diagram illustrating a module to determineapproximate amplitude for indexing the predistortion look-up tables. Itis an embodiment of method 9 from Table 1. It can be implemented withnine shift operations, six additions and two multiplexers.

System Embodiment

FIG. 12 is a block diagram of mobile cellular phone 1000 for use in thenetwork of FIG. 1. Digital baseband (DBB) unit 1002 can include adigital processing processor system (DSP) that includes embedded memoryand security features. Stimulus Processing (SP) unit 1004 receives avoice data stream from handset microphone 1013 a and sends a voice datastream to handset mono speaker 1013 b. SP unit 1004 also receives avoice data stream from microphone 1014 a and sends a voice data streamto mono headset 1014 b. Usually, SP and DBB are separate ICs. In mostembodiments, SP does not embed a programmable processor core, butperforms processing based on configuration of audio paths, filters,gains, etc being setup by software running on the DBB. In an alternateembodiment, SP processing is performed on the same processor thatperforms DBB processing. In another embodiment, a separate DSP or othertype of processor performs SP processing.

RF transceiver 1006 is a digital radio processor as and includes areceiver for receiving a stream of coded data frames from a cellularbase station via antenna 1007 and a transmitter for transmitting astream of coded data frames to the cellular base station via antenna1007. As described in more detail above, at the heart of transceiver1006 lies a digitally controlled oscillator (DCO), which deliberatelyavoids any analog tuning controls. Fine frequency resolution is achievedthrough high-speed dithering of its varactors. Digital logic builtaround the DCO realizes an interpolative all-digital PLL (ADPLL) that isused as a local oscillator for both the transmitter and receiver andoperates as described above. The Cartesian transmitter architectureutilizes the wideband direct frequency modulation capability of theADPLL and a digitally controlled power amplifier (DPA) for the powerramp and amplitude modulation. In this embodiment, a single transceiversupports both GSM and WCDMA operation but other embodiments may usemultiple transceivers for different transmission standards. Otherembodiments may have transceivers for a later developed transmissionstandard with appropriate configuration. RF transceiver 1006 isconnected to DBB 1002 which provides processing of the frames of encodeddata being received and transmitted by cell phone 1000.

The basic WCDMA DSP radio consists of control and data channels, rakeenergy correlations, path selection, rake decoding, and radio feedback.Interference estimation and path selection is performed by instructionsstored in memory 1012 and executed by DBB 1002 in response to signalsreceived by transceiver 1006. Programmable features of the ADPLL withintransceiver 1006 are controlled by instructions executed by DBB 1002.

Transceiver 1006 includes an embodiment of the present invention toperform adaptive gain/phase normalization and adaptive complex gainpredistortion, as described in more detail above. In this embodiment ofthe invention, a least means squared (LMS) based adaptation technique isused for a complex gain lookup table (LUT) to track the temperaturevariations of the transmitter's nonlinear characteristics duringoperation. The algorithm is optimized to enable a low complexityhardware implementation and provides relatively fast convergence. Anamplitude approximation method that is suitable to address the complexgain LUT is embodied that avoids the computation of square root. Also,an adaptive loop gain/phase normalization technique is embodied thatavoids the appearance of curve discontinuities and maintains theaccuracy of open loop power control. It reuses the existing LUTadaptation hardware and therefore is inexpensive to implement.

DBB unit 1002 may send or receive data to various devices connected touniversal serial bus (USB) port 1026. DBB 1002 can be connected tosubscriber identity module (SIM) card 1010 and stores and retrievesinformation used for making calls via the cellular system. DBB 1002 canalso connected to memory 1012 that augments the onboard memory and isused for various processing needs. DBB 1002 can be connected toBluetooth baseband unit 1030 for wireless connection to a microphone1032 a and headset 1032 b for sending and receiving voice data. DBB 1002can also be connected to display 1020 and can send information to it forinteraction with a user of the mobile UE 1000 during a call process.Display 1020 may also display pictures received from the network, from alocal camera 1028, or from other sources such as USB 1026. DBB 1002 mayalso send a video stream to display 1020 that is received from varioussources such as the cellular network via RF transceiver 1006 or camera1028. DBB 1002 may also send a video stream to an external video displayunit via encoder 1022 over composite output terminal 1024. Encoder unit1022 can provide encoding according to PAL/SECAM/NTSC video standards.In some embodiments, audio codec 1009 receives an audio stream from FMRadio tuner 1008 and sends an audio stream to stereo headset 1016 and/orstereo speakers 1018. In other embodiments, there may be other sourcesof an audio stream, such a compact disc (CD) player, a solid statememory module, etc

Other Embodiments

While the invention has been described with reference to illustrativeembodiments, this description is not intended to be construed in alimiting sense. Various other embodiments of the invention will beapparent to persons skilled in the art upon reference to thisdescription. This invention applies to all scheduled communicationsystems which use digitally controlled oscillators. This inventionapplies in uplink and downlink. Various embodiments of this inventionapply for many modulation strategies, which include but are not limitedto, OFDMA, CDMA, DFT-spread FDMA, SC-OFDMA, and others. Embodiments ofthis invention can be applied in most if not all emerging wirelessstandards, including EUTRA.

While a mobile user equipment device has been described, embodiments ofthe invention are not limited to mobile devices. Desktop equipment andother stationary equipment being served by a cellular network may alsoembody an ADPLL as described herein.

Although the invention finds particular application to Digital SignalProcessors (DSPs), implemented, for example, in an Application SpecificIntegrated Circuit (ASIC), it also finds application to other forms ofprocessors. An ASIC may contain one or more megacells which each includecustom designed functional circuits combined with pre-designedfunctional circuits provided by a design library.

An embodiment of the invention may include a system with a processorcoupled to a computer readable medium in which a software program isstored that contains instructions that when executed by the processorperform the functions of modules and circuits described herein. Thecomputer readable medium may be memory storage such as dynamic randomaccess memory (DRAM), static RAM (SRAM), read only memory (ROM),Programmable ROM (PROM), erasable PROM (EPROM) or other similar types ofmemory. The computer readable media may also be in the form of magnetic,optical, semiconductor or other types of discs or other portable memorydevices that can be used to distribute the software for downloading to asystem for execution by a processor. The computer readable media mayalso be in the form of magnetic, optical, semiconductor or other typesof disc unit coupled to a system that can store the software fordownloading or for direct execution by a processor.

As used herein, the terms “applied,” “connected,” and “connection” meanelectrically connected, including where additional elements may be inthe electrical connection path. “Associated” means a controllingrelationship, such as a memory resource that is controlled by anassociated port. The terms assert, assertion, de-assert, de-assertion,negate and negation are used to avoid confusion when dealing with amixture of active high and active low signals. Assert and assertion areused to indicate that a signal is rendered active, or logically true.De-assert, de-assertion, negate, and negation are used to indicate thata signal is rendered inactive, or logically false.

It is therefore contemplated that the appended claims will cover anysuch modifications of the embodiments as fall within the true scope andspirit of the invention.

1. A system comprising a Cartesian transmitter, wherein the transmittercomprises: a transmit chain configured to receive an input signal Xhaving in-phase and quadrature components and having a pre-distorterconfigured to employ a first compensation lookup table operable to holdcomplex valued entries to carry out in-phase and quadraturepre-distortion with respect to the input signal to form a pre-distortedsignal Z, and a nonlinear element configured to process pre-distortedsignal Z to form an output signal Y; a receiver coupled to receiveoutput signal Y and operable to provide a feedback signal in a feedbackloop; an adaptive normalization mechanism operable receive the feedbacksignal and operable to produce an adaptively normalized feed back signal{tilde over (Y)}, using an adaptively updated complex gain normalizationparameter; and a pre-distortion update module operable to update the atleast one compensation lookup table based on the pre-distorted signal Zand the adaptively normalized feedback signal {tilde over (Y)}.
 2. Thesystem of claim 1, wherein the pre-distortion update module comprises: asecond compensation lookup table operable to hold complex valued entriesto carry out in-phase and quadrature post-distortion with respect to theadaptively normalized feedback signal {tilde over (Y)} signal to form apost-distorted feedback signal Zfb, and an error detection module fordetermining a difference in real and imaginary parts between a delayedsample of pre-distorted signal Z and a corresponding sample ofpost-distorted feedback signal Zfb.
 3. The system of claim 2, whereinthe adaptive normalization mechanism is operable to temporarilysubstitute a single register for the second compensation lookup tablewhile updating the complex gain normalization parameter; and wherein theadaptive normalization mechanism is operable to iteratively converge ona revised gain normalization parameter using the single register todetermine a complex value equal to approximately the inverse of acombined gain of the non-linear device over a linear region andcomponents of the feedback loop, such that a resulting loop gain isapproximately one.
 4. The system of claim 3, wherein the adaptivenormalization mechanism is further operable to multiply the revised gainnormalization value by a scale factor to produce the updated complexgain normalization parameter.
 5. The system of claim 1, wherein thetransmit chain further comprises a combiner configured to combine anentry from the first compensation lookup table with the input signal X,wherein the entry is indexed by a representation of the input signal X.6. The system of claim 1, wherein; the transmit chain further comprisesan approximation module to produce an amplitude approximation value foreach sample of the input signal X, wherein the first compensation lookuptable is indexed by the amplitude approximation value; and thepre-distortion compensation module further comprises an approximationmodule to produce an amplitude approximation value for each sample ofthe normalized feed back signal {tilde over (Y)}, wherein the firstcompensation lookup table is indexed by the amplitude approximationvalue.
 7. The system of claim 6, wherein the approximation module isoperable to determine an approximate amplitude of a complex samplehaving a real portion, I and an imaginary portion Q according to:$R = \{ \begin{matrix}{{a_{1}X} + {b_{1}Y}} & {0 \leq \theta < \theta_{B}} \\{{a_{2}X} + {b_{2}Y}} & {\theta_{B} \leq \theta < \frac{\pi}{4}}\end{matrix} $ whereX=max(|I|,|Q|)Y=min(|I|,|Q|) a₁, a₂, b₁, b₂, and θ are selected parameters.
 8. Thesystem of claim 2, wherein an entry LUT[n] in the lookup table selectedin response to the adaptively normalized feedback signal {tilde over(Y)} is updated according to $\begin{matrix}\begin{matrix}{{L\; U\;{T\lbrack n\rbrack}} = {{L\; U\;{T\lbrack n\rbrack}} - {\frac{\mu}{2} \times \frac{{\partial E}\lfloor {ɛ_{k}}^{2} \rfloor}{{\partial L}\; U\;{T\lbrack n\rbrack}}}}} \\{\approx {{L\; U\;{T\lbrack n\rbrack}} + {\mu \times {\overset{\sim}{Y}}^{*} \times ɛ_{k}}}}\end{matrix} & \;\end{matrix}$ where: LUT[n] indicates an entry in the secondcompensation lookup table, {tilde over (Y)}* is a complex conjugatevalue of the normalized feedback signal {tilde over (Y)}, ε_(k) is anerror signal corresponding to a difference between pre-distorted signalZ and post-distorted signal Zfb; and μ is an update factor (real-valued)that controls the speed of convergence.
 9. The system of claim 2,wherein an entry LUT[n] in the lookup table selected in response to theadaptively normalized feedback signal {tilde over (Y)} is updatedaccording toLUT[n]=LUT[n]+δ _(G) where: $\begin{matrix}{\delta_{G} = {ɛ_{k} \times ( {{{sign}( I_{o} )} - {j \cdot {{sign}( Q_{o} )}}} ) \times \mu}} \\{= {ɛ_{k} \times ( {{\pm 1} \pm j} ) \times \mu}} \\{= {ɛ_{k} \times \sqrt{2 \times}{\mathbb{e}}^{{- j}\; k\frac{\pi}{4}} \times \mu}}\end{matrix}$ where k is the quadrant number containing a vector(I_(o)+jQ_(o)) corresponding to normalized feedback signal {tilde over(Y)}, and μ is an update factor (real-valued) that controls the speed ofconvergence.
 10. The system of claim 9, wherein an entry LUT[n] in thelookup table is selected by determining an approximate amplitude of acomplex sample value of signal {tilde over (Y)}.
 11. The system of claim1 being a cellular phone comprising the transmit chain.
 12. A method fortransmitting Cartesian symbols, comprising: pre-distorting an inputsignal X having in-phase and quadrature components using a firstcompensation lookup table operable to hold complex valued entries tocarry out in-phase and quadrature compensation pre-distortion withrespect to the input signal to form a pre-distorted signal Z; processingpre-distorted signal Z to form an output signal Y using a nonlinearelement; adaptively updating a complex gain normalization parameter toreflect varying gain of a linear region of the nonlinear element;forming a normalized feed back signal {tilde over (Y)} using theadaptively updated complex gain normalization parameter; and updatingthe first compensation lookup table based on the pre-distorted inputsignal Z and the adaptively normalized feedback signal {tilde over (Y)}.13. The method of claim 12 wherein updating the first compensation tablecomprises: post-distorting the adaptively normalized feedback signal{tilde over (Y)} signal in a feedback loop to form a post-distortedfeedback signal Zfb using a second compensation lookup table operable tohold complex valued entries to carry out in-phase and quadraturepost-distortion with respect to the adaptively normalized feedbacksignal {tilde over (Y)} signal; determining an error difference in realand imaginary parts between a delayed sample of pre-distorted signal Zand a corresponding sample of post-distorted feedback signal Zfb;updating an entry in the second compensation lookup table with a complexvalue that reduces the error difference; and exchanging the secondcompensation lookup table and the first compensation lookup table. 14.The method of claim 13, wherein adaptively updating the complex gainnormalization parameter comprises: temporarily substituting a singleregister for the second compensation lookup table; and iterativelyconverging on a revised gain normalization parameter using the singleregister to determine a complex value equal to approximately the inverseof a combined gain of the non-linear device over a linear region andcomponents of the feedback loop, such that a resulting loop gain isapproximately one.
 15. The method of claim 14, further comprisingmultiplying the revised gain normalization value by a scale factor toproduce the updated complex gain normalization parameter.
 16. The methodof claim 13, wherein the first compensation lookup table is indexed bythe amplitude approximation value of the input signal X; and wherein thesecond compensation lookup table is indexed by the amplitudeapproximation value of the adaptively normalized feedback signal {tildeover (Y)}.
 17. The method of claim 16, further comprising determining anapproximate amplitude of a complex sample having a real portion, I andan imaginary portion Q according to: $R = \{ \begin{matrix}{{a_{1}X} + {b_{1}Y}} & {0 \leq \theta < \theta_{B}} \\{{a_{2}X} + {b_{2}Y}} & {\theta_{B} \leq \theta < \frac{\pi}{4}}\end{matrix} $ where:X=max(|I|,|Q|)Y=min(|I|,|Q|) a₁, a₂, b₁, b₂, and θ are selected parameters.
 18. Themethod of claim 13, wherein an entry LUT[n] in the lookup table selectedin response to the adaptively normalized feedback signal {tilde over(Y)} is updated according to $\begin{matrix}\begin{matrix}{{L\; U\;{T\lbrack n\rbrack}} = {{L\; U\;{T\lbrack n\rbrack}} - {\frac{\mu}{2} \times \frac{{\partial E}\lfloor {ɛ_{k}}^{2} \rfloor}{{\partial L}\; U\;{T\lbrack n\rbrack}}}}} \\{\approx {{L\; U\;{T\lbrack n\rbrack}} + {\mu \times {\overset{\sim}{Y}}^{*} \times ɛ_{k}}}}\end{matrix} & \;\end{matrix}$ where: LUT[n] indicates an entry in the secondcompensation lookup table, {tilde over (Y)}* is a complex conjugatevalue of the normalized feedback signal {tilde over (Y)}, ε_(k) is anerror signal corresponding to a difference between pre-distorted signalZ and post-distorted signal Zfb, μ is an update factor (real-valued)that controls the speed of convergence.
 19. The method of claim 13,wherein an entry LUT[n] in the lookup table selected in response to theadaptively normalized feedback signal {tilde over (Y)} is updatedaccording toLUT[n]=LUT[n]+δ _(G) where $\begin{matrix}{\delta_{G} = {ɛ_{k} \times ( {{{sign}( I_{o} )} - {j \cdot {{sign}( Q_{o} )}}} ) \times \mu}} \\{= {ɛ_{k} \times ( {{\pm 1} \pm j} ) \times \mu}} \\{= {ɛ_{k} \times \sqrt{2 \times}{\mathbb{e}}^{{- j}\; k\frac{\pi}{4}} \times \mu}}\end{matrix}$ and where k is the quadrant number containing a vector(I_(o)+jQ_(o)) corresponding to normalized feedback signal {tilde over(Y)}.
 20. The method of claim 19, wherein an entry LUT[n] in the lookuptable is selected by determining an approximate amplitude of a complexsample value of signal {tilde over (Y)}.